Name: Principle of Inclusion and Exclusion
Uploaded: 2019-07-19
Duration: 18 min 3 s
Description: The inclusion-exclusion principle is an important combinatorial way to compute the size of a set or the probability of complex events. It relates the sizes o...

Question

A person writes letters to five friends and addresses on the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that
(a) all letters are in the wrong envelopes?
(b) at least three of them are in the wrong envelopes?

Accepted Answer

Given, a person writes letters to $5$ friends and addresses on the corresponding envelopes.

$(a)$ When all letters are in the wrong envelopes, this is the case derangement so that you have to use derangement formula

$= 5! (1 - \frac{1}{1! 1} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!})$

$= 120 (1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120})$

$= 60 - 20 + 5 - 1$

$= 44 .$

$(b)$ When at least $3$ of them are in the wrong envelopes, this is the case derangement so that you have to use derangement formula,

If $3$ letters are in wrong envelope, then $2$ letters in right envelope.

If $4$ letters are in wrong envelope, then $1$ letter in right envelope.

If $5$ letters are in wrong envelope, then zero letters in right envelope.

$\Rightarrow {}^{5}C_{2} \cdot 3! (1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}) + {}^{5}C_{1} \cdot 4! (1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}) + {}^{5}C_{0} \cdot 5! (1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!})$

$= \frac{5!}{2! \cdot 3!} \cdot (3 - 1) + \frac{5!}{4!} \cdot (12 - 4 + 1) + (60 - 20 + 5 - 1)$

$= 20 + 45 + 44$

$= 109 .$

A person writes letters to five friends and addresses on the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that
(a) all letters are in the wrong envelopes?
(b) at least three of them are in the wrong envelopes?

Important Questions on Permutation and Combination

Learn and Prepare for any exam you want !

A person writes letters to five friends and addresses on the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that (a) all letters are in the wrong envelopes? (b) at least three of them are in the wrong envelopes?

Important Questions on Permutation and Combination

Learn and Prepare for any exam you want !

A person writes letters to five friends and addresses on the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that
(a) all letters are in the wrong envelopes?
(b) at least three of them are in the wrong envelopes?